MATH 5333:
Assignment #2
Section 12
3 Let
s
denote sup
S
, and let
m
denote max
S
.
(a)
s
=
m
= 3.
(b)
s
=
m
=
π
.
(c)
s
=
m
= 4.
(d)
s
= 4,
no max.
(e)
s
=
m
= 1.
(f)
s
= 1,
no max.
(g)
s
= 1,
no max.
(h)
s
=
m
= 3
/
2.
(i) None.
(j)
s
= 4,
no max.
(k)
s
=
m
= 1.
(l)
s
= 2,
no max.
(m)
s
= 5,
no max
(n)
s
=
√
5,
no max.
4 Let
t
denote inf
S
, and let
k
denote min
S
.
(a)
t
=
k
= 1.
(b)
t
=
k
= 3.
(c)
t
=
k
= 0.
(d)
t
= 0, no min.
(e)
t
= 0,
no min.
(f)
t
=
k
= 0.
(g)
t
=
k
= 1
/
2.
(h)
t
=
k
=

2.
(i)
t
=
k
= 0.
(j) Neither exists.
(k)
t
=
k
= 1.
(l)
t
= 0,
no min.
(m) Neither exists.
(n)
t
=

√
5,
no min.
6 (a) Suppose that
α
and
β
are least upper bounds for
S
.
Since
α
= sup
S
and
β
is an upper bound for
S
,
α
≤
β
.
Since
β
= sup
S
and
α
is an upper bound for
S
,
β
≤
α
. Therefore
α
=
β
.
(b) Same proof.
8
S, T
nonempty bounded subsets of
R
and
S
⊆
T
.
Choose any
s
∈
S
.
Then
s
∈
T
.
Therefore
s
≥
inf
T
. It follows that inf
T
is a lower bound for
S
and therefore inf
T
≤
inf
S
.
Similarly,
s
≤
sup
T
and so
sup
T
is an upper bound for
S
. Thus,
sup
S
≤
sup
T
.
10 (a) Between
x
and
y
there is a rational number
r
1
. Between
x
and
r
1
there is a rational
number
r
2
. Between
x
and
r
2
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 Fall '08
 Staff
 Math, Empty set, Order theory, β, α, Closed set, General topology